Strong local homogeneity and coset spaces

We prove that for every homogeneous and strongly locally homogeneous separable metrizable space $X$ there is a metrizable compactification $\gamma X$ of $X$ such that, among other things, for all $x,y\in X$ there is a homeomorphism $f\colon\gamma X\to \gamma X$ such that $f(x)=y$. This implies that $X$ is a coset space of some separable metrizable topological group $G$.